Thermodynamic Evidence for a Two-Component Superconducting Order Parameter in Sr2RuO4 Sayak Ghosh,1 Arkady Shekhter,2 F. Jerzembeck,3 N. Kikugawa,4 Dmitry A. Sokolov,3 Manuel Brando,3 A. P. Mackenzie,3 Clifford W. Hicks,3 and B. J. Ramshaw∗1 1Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853, USA 2National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA 3Max Planck Institute for Chemical Physics of Solids, Dresden, Germany 4National Institute for Materials Science, Tsukuba, Ibaraki 305-0003, Japan arXiv:2002.06130v2 [cond-mat.supr-con] 23 Sep 2020 2 Sr2RuO4 has stood as the leading candidate for a spin-triplet superconductor for 26 years. Recent NMR experiments have cast doubt on this candidacy, however, and it is difficult to find a theory of superconductivity that is consistent with all experiments. What is needed are symmetry-based experiments that can rule out broad classes of possible superconducting order parameters. Here we use resonant ultrasound spectroscopy to measure the entire symmetry-resolved elastic tensor of Sr2RuO4 through the superconducting transition. We observe a thermodynamic discontinuity in the shear elastic modulus c66, requiring that the superconducting order parameter is two-component. A two-component p- wave order parameter, such as px + ipy, naturally satisfies this requirement. As this order parameter appears to be precluded by recent NMR experiments, we suggest that two other two-component order parameters, namely fdxz; dyzg or dx2−y2 ; gxy(x2−y2) , are now the prime candidates for the order parameter of Sr2RuO4. INTRODUCTION Nearly all known superconductors are \spin-singlet", composed of Cooper pairs that pair spin-up electrons with spin-down electrons. Noting that Sr2RuO4 has similar normal-state properties to superfluid 3He, Rice and Sigrist[1] and, separately, Baskaran [2], suggested that Sr2RuO4 may be a solid-state \spin-triplet" superconductor. This attracted the attention of the experimental community, and ensured decades of intense research on Sr2RuO4 that resulted in a detailed understanding of its metallic state [3, 4]. From this well-understood starting point one might expect the superconductivity of Sr2RuO4 to be a solved problem [5], but decades after its discovery the symmetry of the superconducting order parameter remains a mystery, largely due to discrepancies between several major pieces of experimental evidence [6]. Formerly, the strongest evidence for spin-triplet pairing in Sr2RuO4 was a Knight shift that was unchanged upon entering the superconducting state [7]. A recently revised version of this experiment, however, shows that the Knight shift is suppressed below Tc, ruling against most spin-triplet order parameters [8, 9]. This is consistent with measurements of the upper-critical magnetic field, which appears to be Pauli-limited and thus suggests spin- 3 singlet pairing [10]. The challenge is to reconcile these data with previous evidence in favour of a spin-triplet order parameter, including time-reversal symmetry breaking below Tc in µSR[11] and polar Kerr effect experiments[12], and half-quantized vortices [13]. While the spin-triplet versus spin-singlet aspect of the superconductivity in Sr2RuO4 is still under debate, less well studied is the symmetry of the orbital part of the Cooper pair wavefunction. By symmetry, spin-triplet superconductors are required to have an odd- parity orbital wavefunction, i.e. to be an l = 1 `p-wave' or l = 3 `f-wave' superconductor, where l is the orbital quantum number. This is in contrast with conventional l = 0 s-wave superconductors, or the high-Tc l = 2 d-wave superconductors. While some information about the orbital wavefunction can be inferred by looking for nodes in the superconducting gap, determination of nodal position does not uniquely determine the orbital structure of the Cooper pair. One way to distinguish different orbital states is by their degeneracy|the number of states with the same energy. s-wave and dx2−y2 -wave Cooper pairing states, for example, are both singly degenerate (\one-component"), while the fpx; pyg state (which can order in the chiral px + ipy configuration) is two-fold degenerate (\two-component"). This dif- ference in orbital degeneracy has an unambiguous signature in an ultrasound experiment: shear elastic moduli are continuous through Tc for a singly-degenerate orbital state, but are discontinuous across Tc for a doubly degenerate state [14, 15]. The observation of a discon- tinuity in one of the shear elastic moduli of Sr2RuO4 at Tc would therefore constitute strong evidence in favour of either p-wave superconductivity, or one of the other two-component superconducting order parameters. These measurements have been attempted in the past, and were suggestive of a shear discontinuity at Tc, but a discontinuity was also found in a symmetry-forbidden channel, and the experiment was thus deemed to be inconclusive [16]. Other independent evidence of a shear discontinuity [17] is now being submitted as part of a separate complementary study using an experimental technique different from our own (for a theoretical interpretation of these results, see Walker and Contreras [18]). EXPERIMENT Elastic moduli are second derivatives with respect to strain of a system's total free energy. Elastic moduli are therefore thermodynamic coefficients akin to heat capacity or magnetic 4 Compression Shear Ru Sr O Representation A1g B1g B2g Eg Elastic modulus /2 c33 c66 c44 r e t One-component o e t e.g. “d-wave” m g � .η2 a ~ A1g ✖ ✖ ✖ n η d r x² - y² i l a p p Two-component u r o e e.g. “p-wave” C d � η2 + η2 � η2 η2 � η η r ~ A .( � y ) B .( � y ) B .( � y) ✖ η (px, py) 1g 1g 2g o � � FIG. 1. Irreducible strains in Sr2RuO4 and their coupling to superconducting order parameters. The tetragonal crystal structure of Sr2RuO4 and unit cell deformations illustrating the irreducible representations of strain are shown. There is an elastic modulus corresponding to each of these strains, and a sixth modulus c13 that arises from coupling between the two A1g strains. Green check marks denote allowed linear-order couplings to strain for one and two-component order parameter bilinears, and red crosses denote that such coupling is forbidden. These couplings are what lead to discontinuities in the elastic moduli at Tc. See Table I for a list of relevant possible order parameters in Sr2RuO4. susceptibility, and are indicative of a system's ground-state properties. Strain is a second- rank tensor quantity, and thus it can couple to order parameters in ways that lower-rank quantities, such as temperature and electric field, cannot. This, in particular, requires that elastic moduli behave differently in systems with one- or two-component order parameters. Here we provide a brief overview of the connection between crystal symmetry, order param- eter symmetry, and ultrasound: the detailed derivations can be found in the `Strain-Order Parameter Coupling' section in S.I., as well as in a number of theoretical papers [14, 18, 19]. The allowed couplings between strains and superconducting order parameters become transparent when both are described in terms of irreducible representations (irreps) of the 5 (a) (b) (c) Max (d) (e) Displacement Min (f) 1.5 b ) c mV ( 1.0 0.5 e d Amplitude 0.0 2.4 2.5 2.6 2.7 2.8 2.9 Frequency (MHz) FIG. 2. Resonant ultrasound spectroscopy: schematic and spectrum. (a) A single-crystal sample, polished along known crystal axes, is held in weak-coupling contact between two ultrasonic transducers, allowing it to vibrate freely at its resonance frequencies. Panels (b) through (e) show the crystal's deformation corresponding to four particular experimentally measured resonances, marked in (f). (f) A portion of the ultrasonic spectrum of Sr2RuO4, in the frequency range from 2.4-2.9 MHz, taken at room temperature. Each resonance creates a unique strain pattern in the material that can be decomposed in terms of the five irreducible strains (Figure 1(a)), modulated in phase along the dimensions of the sample. point-group symmetry. Sr2RuO4 crystallizes in the tetragonal space group I4=mmm, along with its associated point group D4h. In this crystal field environment, the five-component l = 2 d-representation breaks into three one-component irreps: dz2 (A1g irrep), dxy (B2g), and 6 dx2−y2 (B1g|the familiar `d-wave' of the cuprates), and one two-component irrep fdxz; dyzg (Eg). The three-component p-representation breaks into the one-component irrep pz (A2u) and the two-component irrep fpx; pyg (Eu)|it is this latter representation that has been proposed to order into the chiral px + ipy superconducting state. As illustrated in Figure 1, there are five unique strains (Γ) in Sr2RuO4 (five irreps (Γ) of strain in D4h): two compressive strains transforming as the A1g irrep, and three shear strains transforming as the B1g, B2g and Eg irrep. Each strain has a corresponding elastic modulus, 2 2 cΓ = @ F=@Γ, where F is the thermodynamic free energy. A sixth modulus, c13, defines the coupling between the two A1g strains (xx + yy and zz.) Sound velocities can be computed p from these moduli as vΓ = cΓ/ρ, where ρ is the density. When composing terms in the free energy, direct (linear) coupling between strain and the superconducting order parameter (η) is forbidden because superconductivity breaks gauge symmetry. The next relevant coupling is linear in strain and quadratic in order parameter. For one-component superconducting order parameters, including all s-wave states and the dx2−y2 state, the only quadratic form is 2 2 η , transforming as A1g, and thus the only allowed coupling is A1g η . This coupling produces discontinuities in all the A1g (compressional) elastic moduli across Tc. Two-component or- der parameters (~η = fηx; ηyg), on the other hand, have three independent quadratic forms: 2 2 2 2 ηx + ηy, ηx − ηy, and ηxηy, transforming as A1g, B1g, and B2g, respectively.